OF ROTATING COMPRESSIBLE MASSES. 
171 
so that (cf. § 8) 
P = L^ 4 + Mi ? 4 + Nf 4 + 2^ 2 + 2m^ 2 +2nfV+2p^ 2 + 2^ 2 +2r^' 2 . . . (56) 
As far as terms of first degree in e, we may, from equation (12), take 
^(g) = 4P-i/DP+g l 4/ 2 D 2 P],.(57) 
the remaining terms disappearing because P contains no terms of degree higher than 
four in £ >/, £ The whole potential of the solid of unit density, as far as terms in e, 
is from formula (14) 
V, (1) + e A V, (1) = - f if -+ * (:1)] ~■■ 
Jo 
so that 
AV,(1) = -7T ahc [P—i/DP + -<hf 2 P 2 P] 
d\ 
in which q is put equal to 1. 
Equation (51) now becomes 
(58) 
f [P-i/DP+A/TOjA-eP, 
J 0 
= i(2J^ 4 +2ZJ BC ?/V-J)-|(y-2)0(2^) +A >i(x 2 + y 2 ). . . (59) 
\ CL j 
Clearly the left-hand member of this equation will be a linear function of L, M, N, 
while the right-hand member does not involve these coefficients. Thus the values of 
L, M, "N, ... will each be the sum of a number of contributions corresponding to the 
different terms on the right of the equation. Let us suppose that 
P« = P' 0 +P",(y-2),.(60) 
and that 
L = L' + L" (y —2), &c.,.(61) 
thus separating the contribution from the term in (y —2) from the remainder. 
14. It will be remembered that e AV t - (l) is the increase of internal potential 
resulting from deforming the surface of an ellipsoid of unit density so that its surface 
is changed from 
x 2 /a 2 + y 2 /b 2 +z 2 /c 2 -l =0 , . 
into 
.r 2 /a 2 + y 2 jb 2 + z 2 Jc 2 — 1 + eP„ = 0 
Consider the special deformation in which P„ is of the form 
(62) 
(63) 
